Generator Matrix
Given a linear code C, a generator matrix G of C is a matrix whose rows generate all the elements of C, i.e., if G=(g_1 g_2 ... g_k)^(T), then every codeword w of C can be represented as
| w=c_1g_1+c_2g_2+...+c_kg_k=cG |
in a unique way, where c=(c_1 c_2 ... c_k).
An example of a generator matrix is the Golay code, which consists of all 2^(12) possible binary sums of the 11 rows.
See also
Coding Theory, Error-Correcting Code, Linear Code, Parity Check MatrixPortions of this entry contributed by David Terr
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References
Roman, S. Coding and Information Theory. New York: Springer-Verlag, 1992.van Lint, J. H. An Introduction to Coding Theory, 2nd ed. New York: Springer-Verlag, 1992.Referenced on Wolfram|Alpha
Generator MatrixCite this as:
Terr, David and Weisstein, Eric W. "Generator Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneratorMatrix.html